GENERALIZED η-DUALS OF BANACH SPACE VALUED DIFFERENCE SEQUENCE SPACES
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In the present paper, we get an opportunity to introduce and study the notion of generalized η-dual for Banach space valued difference sequence spaces, as a generalization of the classical α-Köthe Toeplitz dual for scalar sequences. We obtain a set of necessary and sufficient conditions for ( 𝐴 𝑘 ) ∈ 𝐸 𝜂 ( 𝑋 , Δ ) (A k )∈E η (X,Δ), where 𝐸 ∈ { ℓ ∞ , 𝑐 , 𝑐 0 } E∈{ℓ ∞ ,c,c 0 }. Moreover, we explore the notion of generalized η-dual for generalized difference sequence spaces 𝐸 ( 𝑋 , Δ 𝑟 ) E(X,Δ r ) and 𝐸 ( 𝑋 , Δ 𝜈 ) E(X,Δ ν ), where 𝑟 ∈ N r∈N and 𝜈 ν is a multiplier sequence.
Abstract
In the present paper, we get an opportunity to introduce and study the notion of generalized η-dual for Banach space valued difference sequence spaces, as a generalization of the classical α-Köthe Toeplitz dual for scalar sequences. We obtain a set of necessary and sufficient conditions for ( 𝐴 𝑘 ) ∈ 𝐸 𝜂 ( 𝑋 , Δ ) (A k )∈E η (X,Δ), where 𝐸 ∈ { ℓ ∞ , 𝑐 , 𝑐 0 } E∈{ℓ ∞ ,c,c 0 }. Moreover, we explore the notion of generalized η-dual for generalized difference sequence spaces 𝐸 ( 𝑋 , Δ 𝑟 ) E(X,Δ r ) and 𝐸 ( 𝑋 , Δ 𝜈 ) E(X,Δ ν ), where 𝑟 ∈ N r∈N and 𝜈 ν is a multiplier sequence.
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